Is this criterion for sequence convergence true?

Question

I am studying sequences in $\mathbb{R}$. I would like to know if a sequence satisfies that "For all $n \in \mathbb{N}, |a_{n+1} - a_{n}| \leq \frac{1}{3^{n}}$" then sequence $\left\{a_{n}\right\}_{n \in \mathbb{N}}$ is convergent is true. I have looked for lots of examples and I have the same conditions for convergent sequences. Any counterexample?

Answer 1

It's not true.

A counterexample is $a_n=\sqrt n$.

Answer 2

Another classic counterexample is the Harmonic Series.